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views by quoting the opinion of M. Biot, that were the

discovery of Leibnitz to be made even now, it would be considered a surprising creation, and must still be acquired, supposing nothing more than the method of Newton, as it is contained in his works, existed. This is not precisely the same as saying that the two things are dissimilar,' as Dr Guhrauer boldly affirms; neither do the assertions of Euler, Lagrange, Laplace, Poisson, also referred to by him, amount to as much.

We do not think that the advocates of Leibnitz wisely consult his fame, by advancing claims that certainly are not tenable. To whatever point of perfection beyond Newton, Leibnitz may have carried his Calculus, we need not hesitate to say, that a decision as to whether the two methods be essentially the same or not, may be regarded as a test of controversial candour or perverseness. Any one competently acquainted with both, and not afflicted with polemical strabismus, would as soon affirm, that German printed in the German type was a different language from German printed in the Roman type, as affirm that the methods of Fluxions and of Differences were essentially distinct things; or he would as soon affirm that two systems of Stenography, each employing the same principles of abbreviation, and differing only in the characters, were essentially different. Whether Leibnitz was truly an independent inventor of this method—in principle identical with that of Fluxions—is the only question, in our judgment, that really affects his fair fame; and that he was so, is now, we may, say, all but universally regarded as indisputable. Involved and complicated as the question has been through the passions and prejudices of contemporary controvertists, its solution really depends upon one very simple question, which we are in a much better position to answer fairly than the heated disputants of that age. It is this,-Was Leibnitz capable of committing the vilest literary larceny, and persisting, to conceal it, in a detestably mean and deliberate falsehood? If not, (and there are few but will acquit him, who consider the general frankness and open. ness of his nature, the freedom with which he communicated his own discoveries, and the candour with which he congratulated others on theirs,) he is entitled to the honours of independent invention. If he was capable of such conduct, then no evidence can satisfy the doubter; for there was assuredly one period during which there was a possibility of deriving advan

* See some excellent remarks on this subject, in Professor De Morgan’s Differential and Integral Calculus, p. 32-34.

tage from the previous discovery of Newton.

The matter stands briefly thus. In the year 1666, Newton, when but twenty-four years of age, was already in possession of the system of Fluxions. Either wishing to exhibit his method in a more perfect form than he had then leisure to impart to it, or desirous of reserving his discovery for his own exclusive benefit, he did not publish it,—though he communicated the outlines of it to some of his friends, and, amongst the rest, to Dr Barrow. The Papers were lent by Barrow to Mr Collins, who, unknown to Newton, took a copy of them, and who showed them to Oldenburg; and as these gentlemen, to use the language of the Royal Society, were very free in communicating to Continental Mathematicians what was going on at home;-as the latter was certainly in communication with

Leibnitz as early as the year 1673, when he visited England; and lastly, as both of them saw him in his subsequent visit in 1676, it has been surmised that Leibnitz might thus have either obtained a glance of these Papers, or some significant hints as to their contents. Now this is precisely the weak point in Leibnitz's case; but we venture to say, that it ought not to weigh against the repeated protestations with which he affirms that he had derived no such advantage; and that he was absolutely ignorant of the name, notation, and nature of Newton's system till some time after 1684, when he published his own first exposition of his Calculus. He repeatedly makes this statement; and, amongst other places, in his correspondence with the Abbé Conti, who was anxious to reconcile the angry disputants. It was precisely this charge against his honour, implied in the statement of Dr Keill, of which Leibnitz most bitterly complains.

There is one part of the statement just alluded to, and it is virtually justified in the well-known Report of the Committee of the Royal Society appointed to investigate this affair, and which compiled the celebrated collection of papers entitled Commercium Epistolicum, which has always appeared to us not only of little weight, as opposed to the solemn protestations above mentioned, but as palpably illogical. We are not aware that the peculiar infirmity in the argument to which we now refer, has ever been exposed, and it may therefore justify us in bestowing a few sentences upon it. As the charge of having possibly seen something explicit on the subject, in the papers, or in the communications of Newton's friends, was but vague, Keill proceeds to say, that the two well-known Letters, which had certainly been communicated to Leibnitz through Mr Oldenburg, contain 'indications of the system of fluxions, sufficiently intelligible to an

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• acute mind, * from which Leibnitz derived, or at least might • derive, the principles of his Calculus.'

The first was communicated in June 1676, and the second in October 1676. In the first, Newton gives an expression for the expansion in serieses of binomial powers; as also expressions for the sine in terms of the arc, for the arc in terms of the sine, &c. &c.; but the Letter contains not a hint of his method of Fluxions. In the second, elicited by a reply from Leibnitz, which clearly showed that the German mathematician was in the track of the same discoveries, Newton details the manner in which he first arrived at his method of Series—its application in 1665 to the quadrature of the hyperbola, and the construction of logarithms; and communicates many other remarkable things,' to use the words of Montucla. But still, results only are given ; no hint is afforded of the methods by which they are attained.

are attained. The method of Fluxions,' says the late eminent Professor Playfair, 'is not communicated • in these letters; nor are the principles of it in any

way suggested.' Nous remarquons ici,' says Montucla,-in reply to the insinuation that the second letter might have given some light,— qu'après avoir lu et relu cette lettre, nous y trou

vons seulement cette méthode décrite quant à ses effets et ses avantages, mais non quant à ses principes.' Those principles Newton conceals in a couple of anagrams, consisting of the transposed letters of the sentences which express them.

Now we affirm that it was in the highest degree unjust and inconsequential to say that Newton had afforded, in documents thus guarded, indications sufficiently intelligible to an acute mind, from which Leibnitz derived, or at least might derive,

the principles of his calculus.' Newton, it is evident, did not think so.

His very object was, whether wisely or unwisely, to keep the matter secret; and it is clear that he thought his reserve and his ciphers would effectually secure that purpose. It is really a species of impertinence, scarcely consistent with the reverence due to Newton's sagacity, to say that what he thought sufficiently guarded was sufficiently intelligible to an . acute mind;' and that, while he flattered himself that he had rendered the matter sufficiently dark, he had, in the very way in which he proposed the enigma, contrived to solve it!

We may be assured he was far more likely than Keill to judge correctly as to what regarded his secret; nor do we believe there is any one, who will calmly read the Letters in question,

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* Keill even goes further,— His indiciis atque his adjutum exemplis, ingenium vulgare methodum Newtonianum penitus discerneret.'Commercium Epistolicum, No. 84.

who will maintain that this great man's sagacity was here at fault. If Leibnitz had really excogitated the differential Calculus out of such materials as these letters, it would have beer scarcely a less illustrious trophy of his genius than the discovery of the Calculus itself; while, if he had been able to make any thing at all of the hieroglyphical ciphers, he must have had no less than the skill of that philosopher in Laputa, who, as Swift tells us, was employed in extracting sunbeams out of cucumbers. In case, however, any tyro in the mathematics should think that these ciphers may have afforded some more hopeful basis of discovery, we give them below.*

In further confirmation of the claims of Leibnitz to the honour of independent discovery, it may be remarked, that though no candid man can deny the essential identity of the two methods, the very differences of terms and notation indicate that they were arrived at by distinct trains of thought, and that the subject was regarded from different points of view. The idea of the generation of magnitudes by the motion of a point, a line, or a surface, was the conception from which Newton worked; Leibnitz, from the idea of magnitudes, as consisting of infinitely small elements, and admitting increase or diminution by infinitely small increments or decrements. · Newton and Leibnitz,' says a candid and competent judge, (Professor De Morgan,) 'had

independently come to the consideration of quantity, and each • made the new step of connecting his ideas with a specific nota« tion.'

It may seem remarkable, that two different men should have made this sublime discovery at the same time, but we must remember, that the necessities of science were simultaneously turning the attention of all the mathematical genius of the age, and even of the preceding one, in the same direction ; and that Newton and Leibnitz were both pre-eminently gifted with powers of invention and analysis. Indeed, so far had previous mathematicians paved the way for the solution of the great problem, that we may well say with Professor De Morgan, • It has, perhaps, not been sufficiently remarked, how nearly

several of their predecessors approached the same ground; ' and it is a question worthy of discussion, whether either • Newton or Leibnitz might not have found broader hints

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in writings accessible to both, than the latter was ever asserted to have received from the former.' *

To conclude merely from the coincidence of their discoveries, that Leibnitz must have stolen from Newton, would be as little reasonable as to suppose that Laplace must have had access to some private sources of information, when, by a very difficult analysis, he proved some of the results which' De Moivre had attained, but which, in accordance with the contracted spirit of the age, the latter simply announced, carrying his methods as a secret to the grave with him.

That Leibnitz was capable of making this discovery, is no such extravagant supposition as to render it necessary to resort to a charge of plagiarism. It is not, perhaps, too much to say, that his mathematical talents were equal to any thing. The masterly manner in which he expounded the principles of the Differential Calculus, and developed its applications, even if we were to suppose its first hints borrowed from Newton; his admirable labours on the Integral Calculus ; the success with which he entered the lists in those intellectual jousts, as they may be called, in which the great mathematicians of the day were wont to engage—the difficult problems he solved, and offered for solution; even his minor achievements—his calculating machine—his binary system of arithmetic—we may add, his juvenile essay De Arte Combinatoria—all show the highly inventive character of his genius, and the subtlety and comprehensiveness of his analytical powers.

If any thing could make us doubt the claims of Leibnitz, it would be a statement of Dr Guhrauer himself-proving, as it would, if true, that Leibnitz was capable of trifling with truth. It is well known that, in 1704, a notice appeared, in the Acta Eruditorum, of Newton's Optics. That notice contained a paragraph, which seemed to imply that Newton had been a plagiarist from Leibnitz. The obnoxious sentence given in all accounts of the controversy was as follows :- Pro differentiis

igitur Leibnitianis D. Newtonus adhibet, semperque adhibuit, • fluxiones; ..... quemadmodum et honoratus Fabrius, in suả Synopsi Geometricâ motuum progressus Cavallerianæ me6thodo substituit.'

Newton felt highly indignant at this paragraph, as he well might—even supposing that no charge of plagiarism was intended. Leibnitz constantly affirmed in reply, that it could be interpreted into a charge of plagiarism only by a false and malicious glossa



Elementary Illustrations of the Differential and Integral Calculus.

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